Relative Gibbs Energies in Solution through Continuum Models:  Effect of the Loss of Translational Degrees of Freedom in Bimolecular Reactions on Gibbs Energy Barriers

2005 ◽  
Vol 109 (49) ◽  
pp. 23618-23623 ◽  
Author(s):  
Diego Ardura ◽  
Ramón López ◽  
Tomás L. Sordo
Keyword(s):  
Gibbs Energy ◽  
Degrees Of Freedom ◽  
Continuum Models ◽  
Energy Barriers ◽  
Bimolecular Reactions ◽  
Gibbs Energies

A Concurrent Multiscale Method for Coupling Atomistic and Continuum Models at Finite Temperatures

2009 ◽  
Vol 1229 ◽  
Author(s):  
Catalin Picu ◽  
Nithin Mathew
Keyword(s):  
Degrees Of Freedom ◽  
Interface Region ◽  
Continuum Models ◽  
Generalized Langevin Equation ◽  
Mechanical Coupling ◽  
Multi Scale ◽  
Scale Modeling ◽  
Temperature Boundary ◽  
Atomistic Region ◽  
The Continuum

AbstractA concurrent multi-scale modeling method for finite temperature simulation of solids is introduced. The objective is to represent far from equilibrium phenomena using an atomistic model and near equilibrium phenomena using a continuum model, the domain being partitioned in discrete and continuum regions, respectively. An interface sub-domain is defined between the two regions to provide proper coupling between the discrete and continuum models. While in the discrete region the thermal and mechanical processes are intrinsically coupled, in the continuum region they are treated separately. The interface region partitions the energy transferred from the discrete to the continuum into mechanical and thermal components by splitting the phonon spectrum into “low” and “high” frequency ranges. This is achieved by using the generalized Langevin equation as the equation of motion for atoms in the interface region. The threshold frequency is selected such to minimize energy transfer between the mechanical and thermal components. Mechanical coupling is performed by requiring the continuum degrees of freedom (nodes) to follow the averaged motion of the atoms. Thermal coupling is ensured by imposing a flux input to the atomistic region and using a temperature boundary condition for continuum. This makes possible a thermodynamically consistent, bi-directional coupling of the two models.

ON THE POSSIBILITY OF THE POSTTRANSITION STATE CLASSICAL TRAJECTORY PREDICTIONS OF ENERGY DEPOSITION IN THE REACTION PRODUCTS' DEGREES OF FREEDOM

2020 ◽  
Author(s):  
B. I. Loukhovitski ◽  
◽  
A. S. Sharipov ◽  
Keyword(s):  
Energy Distribution ◽  
Degrees Of Freedom ◽  
Energy Deposition ◽  
Classical Trajectory ◽  
Reaction Products ◽  
Dynamic Simulations ◽  
Bimolecular Reactions

The possibility of applying the method of posttransition state classical trajectory dynamic simulations to study the nascent energy distribution among the molecular degrees of freedom of the reaction products on the example of a number of bimolecular reactions is considered.

scholarly journals Analysis of Two-State Folding Using Parabolic Approximation II: Temperature-Dependence

2016 ◽  
Author(s):  
Robert S Sade
Keyword(s):  
Protein Folding ◽  
Gibbs Energy ◽  
Temperature Dependence ◽  
Rate Constants ◽  
Heat Capacities ◽  
State System ◽  
Parabolic Approximation ◽  
Temperature Range ◽  
Gibbs Energies ◽  
Novel Treatment

Equations that govern the temperature-dependence of the rate constants, Gibbs energies, enthalpies, entropies and heat capacities of activation for folding and unfolding of spontaneously-folding fixed two-state systems have been derived using a procedure that treats the denatured and the native conformers as being confined to harmonic Gibbs energy wells. The notion that a two-state system is physically defined only for a set temperature range is introduced. The implications of this novel treatment for protein folding are discussed.

scholarly journals An energy‐momentum scheme for extended continuum models with rotational degrees of freedom

PAMM ◽  
2021 ◽  
Vol 21 (1) ◽  
Author(s):  
Michael Groß ◽  
Julian Dietzsch ◽  
Iniyan Kalaimani
Keyword(s):  
Degrees Of Freedom ◽  
Continuum Models ◽  
Rotational Degrees Of Freedom

scholarly journals Modelling thermodynamic properties of binary Cu–Eu and ternary Al–Cu–Eu melts

2020 ◽  
Vol 8 (2) ◽  
pp. 73-82
Author(s):  
Natalia Kotova ◽  
Natalia Golovata ◽  
Natalia Usenko
Keyword(s):  
Thermodynamic Properties ◽  
Gibbs Energy ◽  
Excess Gibbs Energy ◽  
Model Calculations ◽  
Gibbs Energies ◽  
Excess Gibbs Energies ◽  
Energy Of Mixing ◽  
Gibbs Energy Of Mixing ◽  
Associated Solution Model ◽  
Associated Solution

Model calculations of the whole set of thermodynamic properties of liquid alloys for the binary Cu–Eu and ternary Al–Cu–Eu systems have been performed. Authors used the ideal associated solution model (IAS model) for calculation of the entropies and excess Gibbs energies of mixing for these systems. The binaries were given as the Redlich-Kister polynomials. The thermodynamic properties for the ternary system are described using the Redlich-Kister-Muggianu formalism. A comparison of the surfaces of excess Gibbs energy and entropy of mixing for liquid Al–Cu–Eu alloys at 1350 K demonstrates that the ordering related to the formation of rather strong associates in the Al–Eu system significantly affects the concentration dependence of the excess Gibbs energy of mixing in the liquid phase at this temperature.

Dynamic Continuum Models of Large Platelike Lattice Structures

1991 ◽  
Author(s):  
Usik Lee
Keyword(s):  
Degrees Of Freedom ◽  
Dynamic Properties ◽  
Continuum Models ◽  
Periodic Lattice ◽  
Lattice Structures ◽  
Straightforward Method ◽  
Energy Equivalence ◽  
Mass Matrices ◽  
Equivalent Continuum ◽  
The Continuum

Abstract A rational and straightforward method is introduced for developing continuum models of large platelike periodic lattice structures based on energy equivalence. The procedure for developing continuum plate models involves the use of existing well-defined finite element matrices for the easy calculation of strain and kinetic energies of a repeating cell, from which the reduced stiffness and mass matrices are obtained in terms of continuum degrees-of-freedom defined in this paper. The equivalent continuum plate properties are obtained from the direct comparison of the reduced matrices for continuum plate with those for lattice plate. Free vibration analyses for the continuum and lattice plates are conducted to evaluate the continuum method proposed in this paper. Numerical results show that the present continuum method gives very reliable structural and dynamic properties compared to other well-recognized methods.

Symplectic Geometry in Optics

1994 ◽  
Author(s):  
Antony N. Beris ◽  
Brian J. Edwards
Keyword(s):  
Numerical Simulation ◽  
Degrees Of Freedom ◽  
Symplectic Geometry ◽  
Stokes Equations ◽  
Fundamental Problem ◽  
Navier Stokes ◽  
Continuum Models ◽  
Engineering Practice ◽  
Navier Stokes Equations ◽  
Model Equations

The scope of this book is to address the fundamental problem of modeling transport processes within complex systems, i.e., systems with internal microstructure. The classical engineering approach involves the modeling of the systems as structured continua and the subsequent use of the models in order to derive (if possible) analytical results, exact or approximate. The advent of powerful computers and the promise through parallel processing of even more substantial computational gains in the near future have introduced yet another paragon to the established engineering practice: that of the numerical simulation. Numerical simulation has emerged as a viable alternative to experiments (contrast Computational Fluid Dynamics (CFD) simulations versus wind tunnel experiments); however, the key limitation to a wider application of numerical simulations in engineering practice lies in the reliability of the models (as well as in their simplicity). CFD applications are successful since the Navier/Stokes equations which they employ are quite capable of describing accurately enough the hydrodynamics of air and water. However, as we move our emphasis to materials of such internal complexity as polymer melts, liquid crystals, suspensions, etc., the development of reliable continuum models becomes an increasingly arduous task. The main objective of this treatise is to investigate a more systematic approach through which continuum models may be developed and analyzed. The key issue that the modeler has to cope with is how to construct models which describe more of the underlying physics without, at the same time, becoming excessively complex so that they either require a prohibitively large, experimentally determined number of adjustable parameters (such as current phenomenological theories) or a prohibitively large computational time (such as required for a detailed “brute force” description of the molecular dynamics). It is the thesis of the present work that a lot of effort can be saved if the appropriate formulation is used in deriving model equations, a formulation which is capable of exploiting to a maximum degree the inherent symmetry and consistency of the collective phenomena exhibited by a large number of internal degrees of freedom.

Thermodynamic properties of molybdate ion: reaction cycles and experiments

2015 ◽  
Vol 87 (5) ◽  
pp. 461-476 ◽  
Author(s):  
Heinz Gamsjäger ◽  
Masao Morishita
Keyword(s):  
Thermodynamic Properties ◽  
Gibbs Energy ◽  
Metal Ions ◽  
Enthalpy Of Formation ◽  
Alkaline Earth ◽  
Standard Entropy ◽  
Gibbs Energy Of Formation ◽  
Silver Molybdate ◽  
Gibbs Energies ◽  
Energy Of Formation

AbstractStandard molar quantities of molybdate ion entropy, $S_{\rm{m}}^0,$ enthalpy of formation, ${\Delta _{\rm{f}}}H_m^{\rm{o}},$ and Gibbs energy of formation, ${\Delta _{\rm{f}}}G_{\rm{m}}^{\rm{o}},$ are key data for the thermodynamic properties of molybdenum compounds and complexes, which are at present investigated by an OECD NEA review project. The most reliable method to determine ${\Delta _{\rm{f}}}H_{\rm{m}}^{\rm{o}}$ of molybdate ion and alkali molybdates directly consists in measuring calorimetrically the enthalpy of dissolution of crystallized molybdenum trioxide and anhydrous alkali molybdates in corresponding aqueous alkali metal hydroxide solutions. Solubility equilibria of sparingly soluble alkaline earth molybdates and silver molybdate lead to trustworthy data for ${\Delta _{\rm{f}}}G_{\rm{m}}^{\rm{o}}$ of molybdate ion. Thereby the Gibbs energies of the metal molybdates and the corresponding metal ions are combined with the Gibbs energies of dissolution. As reliable values are available for ${\Delta _{\rm{f}}}G_{\rm{m}}^{\rm{o}}$ of the relevant metal ions the problem reduces to select the best values of solubility constants and ${\Delta _{\rm{f}}}G_{\rm{m}}^{\rm{o}}$ of alkaline earth molybdates and silver molybdate. There are two independent possibilities to achieve the latter task. (1) ${\Delta _{\rm{f}}}H_{\rm{m}}^{\rm{o}}$ for alkaline earth molybdates and silver molybdate have been determined by solution calorimetry. Entropy data of molybdenum have been compiled and evaluated recently. CODATA key values are available for $S_{\rm{m}}^{\rm{o}}$ of the other elements involved. Whereas $S_{\rm{m}}^{\rm{o}}({\rm{CaMo}}{{\rm{O}}_4},{\rm{ cr}})$ is well known since decades, low-temperature heat capacity measurements had to be performed recently, but now reliable values for $S_{\rm{m}}^{\rm{o}}$ of Ag2MoO4(cr), BaMoO4(cr) and SrMoO4(cr) are available. (2) ${\Delta _{\rm{f}}}H_{\rm{m}}^{\rm{o}}({\rm{BaMo}}{{\rm{O}}_4},{\rm{ cr}}),$ for example, can be obtained from high temperature equilibria also, but the result is less accurate than that of the first method. Once Gibbs energy of formation, ${\Delta _{\rm{f}}}G_{\rm{m}}^{\rm{o}},$ and enthalpy of formation, ${\Delta _{\rm{f}}}H_{\rm{m}}^{\rm{o}},$ of molybdate ion are known its standard entropy, $S_{\rm{m}}^{\rm{o}},$ can be calculated.

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